Understanding the Relationships of

Functions&

Systems of Equations

MATH 2RICHARDSON 423

2/16/15

Week overview

• This week’s lesson entails a further look into polynomials and their behavior in a graphical application.• This week as a class we will be able to identify the following:• Review of a Cartesian graph

• Vocabulary Terms• A function

• Definition• Identifying examples

• Class of functiona• Degree of functions• Mathematical Terms• Practice Examples

Work overview continued

• Equations• Definition• Examples of Different Equations• Terms• Practice Examples

• Inequalities• Definition• Delineation of equations and inequalities• Examples of various cases by graphing problems/solution

Parts of a Cartesian (Plane)graph

• Vocabulary terms:• Axis:• Y-axis• Vertical directed line

• X-axis• Horizontal directed line that reads for all points perpendicular

to the y axis.• Coordinate:• Points along axis meniscus found from solutions created from

selected functions/equations•Origin• The center point of the x-axis and y-axis.a

Cartesian (Plane) Graph Illustration

The Coordinates Illustration

As you can see from the illustration, a pair of coordinatesAre made by moving either left to right along the x-axis first. (X,The y values are found by locating the values either up or down parallel to the y-axis. ,Y)

Understanding what is a function?

• What exactly is a function?• Defitinition:

• A function is a special relationship where each input has a single output.• A function is often written as “f(x)” where the x is the input value.

• Example: • F(x) =( x/3) (Orally: “F of x is(=) x divided by 3”• This is a function because each input “X” has a single output “x/3”

• F(3)= ( 3/3)=1• F(15)= (15/3)=5• F(-12)=(-12/3)=-4

Let’s understand the three parts of a function

The input The relationship The OutputThe value to place into the function

The actual equation we are substituting the value for x

The final solution once substitution has been made

We address the input as “F( )”

x2+3x+9 = ______________

F(0) (0)2+3(0)+9 =9F(1) (1)2+3(1)+9 =13

Function Rules

Functions are just equations we just substitute values in to find solutions.Functions follow this saying:

“Not all functions are equations, but rather mathematical relationships.”

Meaning they don’t have to follow the same rules or mathematical applications as equations. Abstractly an equation is a lightly defined statement with some variables that can lead to a definite solution to multiple answers. A function relates only to one variable directly that will result to a set of solutions directly equal.A One to one description of a mathematical relationship of numbers.

Functions follow these rules:

1. “…each element….”1. A function relates each element of a set with exactly one element of

another set (possibly the same set).

2. “….exactly one…”A functioin is single valued. It will not give you back 2 or more

results for the same input.Example: f(2)= 7 OR 9

Vertical Line Test

• So after we find our output values and plot them on our Cartesian plane how are we sure that we are dealing with an actual function instead of a mathematical relationship?• We use a vertical line test. On a graph, the idea of a single valued

means that no vertical line ever crosses more that one value in passing across the final ‘coordinate pairs’. • If it crosses more than once it is a still a valid curve that describes the results,

BUT IT IS NOT A FUNCTION.

Practice Examples on Understanding Functions• Write an equation to represent the function from the following table

of values:X Y

-2 -4

-1 -2

0 0

1 2

2 4

A. Y=-2X B. Y=2X

C. Y=X+1 D. Y=X+2

Practice Example

•Which one of the following relations is NOT a function?

Which one of these graphs does not illustrate a function?

Hint: Use the Vertical Line Test to solve this problem.

Which one of the following graphs is not a function?

UNDERSTANDING DOMAIN AND RANGE

• WHAT IS A DOMAIN?• Outside of the terminology for cyberspace pertaining to an

identification string which constitutes a brand and space for a product by way of html code, a domain is a serious term we use in math to define elements.• The DOMAIN is a set of all the values used to go into a function.• These would be the values located on the “ X-AXIS”

• The RANGE is the output values made from the function. • The output values or solutions from the function would be the values

located by the “Y-AXIS”

Domain and Range Illustration

Practice Example

• A = {-3, -2, -1, 0, 1, 2, 3}

f is a function from A to the set of whole numbers as defined in the following table:

A. The set of Integers B. The set of whole numbers

C. {-3,-2,-1,0,1,2,3} D: {0,1,4,9}

Practice Example

• Which relation is not a function?

A. F(x)=√x B. f=-√x

C. F(x)=+√x D. F(x)=√x -1

Practice Example

•The function f is defined on the real numbers by f(x) =2+x-x2? What is the value of f(-3)?A. -10 B. -4

C. 8 D. 14